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Numerical range

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Aspect of a numerical matrix

In themathematical field oflinear algebra andconvex analysis, thenumerical range orfield of values orWertvorrat orWertevorrat of acomplex $n\times n$ matrixA is the set

W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \neq 0\right\}=\left\{\langle \mathbf {x} ,A\mathbf {x} \rangle \mid \mathbf {x} \in \mathbb {C} ^{n},\ \|\mathbf {x} \|_{2}=1\right\}

where $\mathbf {x} ^{*}$ denotes theconjugate transpose of thevector $\mathbf {x}$ . The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosingx equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosingx equal to the eigenvectors).

Equivalently, the elements of ${\textstyle W(A)}$ are of the form ${\textstyle \operatorname {tr} (AP)}$ , where ${\textstyle P}$ is a Hermitianprojection operator from ${\textstyle \mathbb {C} ^{2}}$ to a one-dimensional subspace.

In engineering, numerical ranges are used as a rough estimate ofeigenvalues ofA. Recently, generalizations of the numerical range are used to studyquantum computing.

A related concept is thenumerical radius, which is the largest absolute value of the numbers in the numerical range, i.e.

r(A)=\sup\{|\lambda |:\lambda \in W(A)\}=\sup _{\|x\|_{2}=1}|\langle \mathbf {x} ,A\mathbf {x} \rangle |.

Properties

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Let sum of sets denote asumset.

General properties

The numerical range is therange of theRayleigh quotient.
(Hausdorff–Toeplitz theorem) The numerical range isconvex andcompact.
$W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}$ for all square matrix $A {\displaystyle A}$ and complex numbers $\alpha$ and $\beta$ . Here $I {\displaystyle I}$ is theidentity matrix.
$W(A)$ is a subset of the closed right half-plane if and only if $A+A^{*}$ is positive semidefinite.
The numerical range $W(\cdot )$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
$W(UAU^{*})=W(A)$ for any unitary $U {\displaystyle U}$ .
$W(A^{*})=W(A)^{*}$ .
If $A {\displaystyle A}$ is Hermitian, then $W(A)$ is on the real line. If $A {\displaystyle A}$ isanti-Hermitian, then $W(A)$ is on the imaginary line.
$W(A)=\{z\}$ if and only if $A=zI$ .
(Sub-additive) $W(A+B)\subseteq W(A)+W(B)$ .
$W(A)$ contains all theeigenvalues of $A {\displaystyle A}$ .
The numerical range of a $2\times 2$ matrix is a filledellipse.
$W(A)$ is a real line segment $[\alpha ,\beta ]$ if and only if $A {\displaystyle A}$ is aHermitian matrix with its smallest and the largest eigenvalues being $\alpha$ and $\beta$ .

Normal matrices

If ${\textstyle A}$ is normal, and ${\textstyle x\in \operatorname {span} (v_{1},\dots ,v_{k})}$ , where ${\textstyle v_{1},\ldots ,v_{k}}$ are eigenvectors of ${\textstyle A}$ corresponding to ${\textstyle \lambda _{1},\ldots ,\lambda _{k}}$ , respectively, then ${\textstyle \langle x,Ax\rangle \in \operatorname {hull} \left(\lambda _{1},\ldots ,\lambda _{k}\right)}$ .
If $A {\displaystyle A}$ is a normal matrix then $W(A)$ is the convex hull of its eigenvalues.
If $\alpha$ is a sharp point on the boundary of $W(A)$ , then $\alpha$ is anormal eigenvalue of $A {\displaystyle A}$ .

Numerical radius

$r(\cdot )$ is aunitarily invariant norm on the space of $n\times n$ matrices.
$r(A)\leq \|A\|_{\operatorname {op} }\leq 2r(A)$ , where $\|\cdot \|_{\operatorname {op} }$ denotes theoperator norm.^[1]^[2]^[3]^[4]
$r(A)=\|A\|_{\operatorname {op} }$ if (but not only if) $A {\displaystyle A}$ is normal.
$r(A^{n})\leq r(A)^{n}$ .

Proofs

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Most of the claims are obvious. Some are not.

General properties

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Proof of (13)

If ${\textstyle A}$ is Hermitian, then it is normal, so it is the convex hull of its eigenvalues, which are all real.

Conversely, assume ${\textstyle W(A)}$ is on the real line. Decompose ${\textstyle A=B+C}$ , where ${\textstyle B}$ is a Hermitian matrix, and ${\textstyle C}$ an anti-Hermitian matrix. Since ${\textstyle W(C)}$ is on the imaginary line, if ${\textstyle C\neq 0}$ , then ${\textstyle W(A)}$ would stray from the real line. Thus ${\textstyle C=0}$ , and ${\textstyle A}$ is Hermitian.

The following proof is due to^[5]

Proof of (12)

The elements of ${\textstyle W(A)}$ are of the form ${\textstyle \operatorname {tr} (AP)}$ , where ${\textstyle P}$ is projection from ${\textstyle \mathbb {C} ^{2}}$ to a one-dimensional subspace.

The space of all one-dimensional subspaces of ${\textstyle \mathbb {C} ^{2}}$ is ${\textstyle \mathbb {P} \mathbb {C} ^{1}}$ , which is a 2-sphere. The image of a 2-sphere under a linear projection is a filled ellipse.

In more detail, such ${\textstyle P}$ are of the form ${\frac {1}{2}}I+{\frac {1}{2}}{\begin{bmatrix}\cos 2\theta &e^{i\phi }\sin 2\theta \\e^{-i\phi }\sin 2\theta &-\cos 2\theta \end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1+z&x+iy\\x-iy&1-z\end{bmatrix}}$ where ${\textstyle x,y,z}$ , satisfying ${\textstyle x^{2}+y^{2}+z^{2}=1}$ , is a point on the unit 2-sphere.

Therefore, the elements of ${\textstyle W(A)}$ , regarded as elements of ${\textstyle \mathbb {R} ^{2}}$ is the composition of two real linear maps ${\textstyle (x,y,z)\mapsto {\frac {1}{2}}{\begin{bmatrix}1+z&x+iy\\x-iy&1-z\end{bmatrix}}}$ and ${\textstyle M\mapsto \operatorname {tr} (AM)}$ , which maps the 2-sphere to a filled ellipse.

Proof of (2)

${\textstyle W(A)}$ is the image of a continuous map ${\textstyle x\mapsto \langle x,Ax\rangle }$ from the $\mathbb {PC} ^{n}$ , so it is compact.

Given two complex nonzero vectors ${\textstyle x,y}$ , let ${\textstyle P_{x},P_{y}}$ be their corresponding Hermitian projectors from ${\textstyle \mathbb {C} ^{n}}$ to their respective spans. Let ${\textstyle P}$ be the Hermitian projector to the span of both. We have that ${\textstyle P^{*}AP}$ is an operator on ${\textstyle \operatorname {Span} (x,y)}$ .

Therefore, the “restricted numerical range” of ${\textstyle P^{*}AP}$ , defined by ${\textstyle \{\operatorname {Tr} (P^{*}APP_{z}):z\in \operatorname {Span} (x,y),z\neq 0\}}$ , is a closed ellipse, according to (12). It is also the case that if ${\textstyle z\in \operatorname {Span} (x,y)}$ is nonzero, then ${\textstyle \operatorname {Tr} (P^{*}APP_{z})=\operatorname {Tr} (APP_{z}P)=\operatorname {Tr} (AP_{z})\in W(A)}$ . Therefore, the restricted numerical range is contained in the full numerical range of ${\textstyle A}$ .

Thus, if ${\textstyle W(A)}$ contains ${\textstyle \operatorname {Tr} (AP_{x}),\operatorname {Tr} (AP_{y})}$ , then it contains a closed ellipse that also contains ${\textstyle \operatorname {Tr} (AP_{x}),\operatorname {Tr} (AP_{y})}$ , so it contains the line segment between them.

Proof of (5)

Let ${\textstyle W}$ satisfy these properties. Let ${\textstyle W_{0}}$ be the original numerical range.

Fix some matrix ${\textstyle A}$ . We show that thesupporting planes of ${\textstyle W(A)}$ and ${\textstyle W_{0}(A)}$ are identical. This would then imply that ${\textstyle W(A)=W_{0}(A)}$ since they are both convex and compact.

By property (4), ${\textstyle W(A)}$ is nonempty. Let ${\textstyle z}$ be a point on the boundary of ${\textstyle W(A)}$ , then we can translate and rotate the complex plane so that the point translates to the origin, and the region ${\textstyle W(A)}$ falls entirely within ${\textstyle \mathbb {C} ^{+}}$ . That is, for some ${\textstyle \phi \in \mathbb {R} }$ , the set ${\textstyle e^{i\phi }(W(A)-z)}$ lies entirely within ${\textstyle \mathbb {C} ^{+}}$ , while for any ${\textstyle t>0}$ , the set ${\textstyle e^{i\phi }(W(A)-z)-tI}$ does not lie entirely in ${\textstyle \mathbb {C} ^{+}}$ .

The two properties of ${\textstyle W}$ then imply that $e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}\succeq 0$ and that inequality is sharp, meaning that ${\textstyle e^{i\phi }(A-z)+e^{-i\phi }(A-z)^{*}}$ has a zero eigenvalue. This is a complete characterization of the supporting planes of ${\textstyle W(A)}$ .

The same argument applies to ${\textstyle W_{0}(A)}$ , so they have the same supporting planes.

Normal matrices

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Proof of (1), (2)

For (2), if ${\textstyle A}$ is normal, then it has a full eigenbasis, so it reduces to (1).

Since ${\textstyle A}$ is normal, by the spectral theorem, there exists a unitary matrix ${\textstyle U}$ such that ${\textstyle A=UDU^{*}}$ , where ${\textstyle D}$ is a diagonal matrix containing the eigenvalues ${\textstyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{n}}$ of ${\textstyle A}$ .

Let ${\textstyle x=c_{1}v_{1}+c_{2}v_{2}+\cdots +c_{k}v_{k}}$ . Using the linearity of the inner product, that ${\textstyle Av_{j}=\lambda _{j}v_{j}}$ , and that ${\textstyle \left\{v_{i}\right\}}$ are orthonormal, we have:

$\langle x,Ax\rangle =\sum _{i,j=1}^{k}c_{i}^{*}c_{j}\left\langle v_{i},\lambda _{j}v_{j}\right\rangle =\sum _{i=1}^{k}\left|c_{i}\right|^{2}\lambda _{i}\in \operatorname {hull} \left(\lambda _{1},\ldots ,\lambda _{k}\right)$

Proof (3)

By affineness of ${\textstyle W}$ , we can translate and rotate the complex plane, so that we reduce to the case where ${\textstyle \partial W(A)}$ has a sharp point at ${\textstyle 0}$ , and that the two supporting planes at that point both make an angle ${\textstyle \phi _{1},\phi _{2}}$ with the imaginary axis, such that ${\textstyle \phi _{1}<\phi _{2},e^{i\phi _{1}}\neq e^{i\phi _{2}}}$ since the point is sharp.

Since ${\textstyle 0\in W(A)}$ , there exists a unit vector ${\textstyle x_{0}}$ such that ${\textstyle x_{0}^{*}Ax_{0}=0}$ .

By general property (4), the numerical range lies in the sectors defined by: $\operatorname {Re} \left(e^{i\theta }\langle x,Ax\rangle \right)\geq 0\quad {\text{for all }}\theta \in [\phi _{1},\phi _{2}]{\text{ and nonzero }}x\in \mathbb {C} ^{n}.$ At ${\textstyle x=x_{0}}$ , the directional derivative in any direction ${\textstyle y}$ must vanish to maintain non-negativity. Specifically:
$\left.{\frac {d}{dt}}\operatorname {Re} \left(e^{i\theta }\langle x_{0}+ty,A(x_{0}+ty)\rangle \right)\right|_{t=0}=0\quad \forall y\in \mathbb {C} ^{n},\theta \in [\phi _{1},\phi _{2}].$ Expanding this derivative:
$\operatorname {Re} \left(e^{i\theta }\left(\langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle \right)\right)=0\quad \forall y\in \mathbb {C} ^{n},\theta \in [\phi _{1},\phi _{2}].$

Since the above holds for all ${\textstyle \theta \in [\phi _{1},\phi _{2}]}$ , we must have: $\langle y,Ax_{0}\rangle +\langle x_{0},Ay\rangle =0\quad \forall y\in \mathbb {C} ^{n}.$

For any ${\textstyle y\in \mathbb {C} ^{n}}$ and ${\textstyle \alpha \in \mathbb {C} }$ , substitute ${\textstyle \alpha y}$ into the equation: $\alpha \langle y,Ax_{0}\rangle +\alpha ^{*}\langle x_{0},Ay\rangle =0.$ Choose ${\textstyle \alpha =1}$ and ${\textstyle \alpha =i}$ , then simplify, we obtain $\langle y,Ax_{0}\rangle =0$ for all $y {\displaystyle y}$ , thus ${\textstyle Ax_{0}=0}$ .

Numerical radius

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Proof of (2)

Let ${\textstyle v=\arg \max _{\|x\|_{2}=1}|\langle x,Ax\rangle |}$ . We have ${\textstyle r(A)=|\langle v,Av\rangle |}$ .

By Cauchy–Schwarz, $|\langle v,Av\rangle |\leq \|v\|_{2}\|Av\|_{2}=\|Av\|_{2}\leq \|A\|_{op}$

For the other one, let ${\textstyle A=B+iC}$ , where ${\textstyle B,C}$ are Hermitian. $\|A\|_{op}\leq \|B\|_{op}+\|C\|_{op}$

Since ${\textstyle W(B)}$ is on the real line, and ${\textstyle W(iC)}$ is on the imaginary line, the extremal points of ${\textstyle W(B),W(iC)}$ appear in ${\textstyle W(A)}$ , shifted, thus both ${\textstyle \|B\|_{op}=r(B)\leq r(A),\|C\|_{op}=r(iC)\leq r(A)}$ .

Generalisations

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Higher-rank numerical range

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The numerical range is equivalent to the following definition: $W(A)=\{\lambda \in \mathbb {C} :PMP=\lambda P{\text{ for some Hermitian projector }}P{\text{ of rank }}1\}$ This allows a generalization tohigher-rank numerical ranges, one for each $k=1,2,3,\dots$ :^[6] $W_{k}(A)=\{\lambda \in \mathbb {C} :PMP=\lambda P{\text{ for some Hermitian projector }}P{\text{ of rank }}k\}$ $W_{k}(A)$ is always closed and convex,^[7]^[8] but it might be empty. It is guaranteed to be nonempty if $k<n/3+1$ , and there exists some $A {\displaystyle A}$ such that $W_{k}(A)$ is empty if $k\geq n/3+1$ .^[9]

Bibliography

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Books

Bonsall, F.F.; Duncan, J. (1971),Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras,Cambridge University Press,ISBN 978-0-521-07988-4
Bonsall, F.F.; Duncan, J. (1973),Numerical Ranges II,Cambridge University Press,ISBN 978-0-521-20227-5
Horn, Roger A.; Johnson, Charles R. (1991),Topics in Matrix Analysis,Cambridge University Press, Chapter 1,ISBN 978-0-521-46713-1.
Horn, Roger A.; Johnson, Charles R. (1990),Matrix Analysis,Cambridge University Press, Ch. 5.7, ex. 21,ISBN 0-521-30586-1
Bhatia, Rajendra (1997).Matrix analysis. Graduate texts in mathematics. New York Berlin Heidelberg: Springer.ISBN 978-0-387-94846-1.
Gustafson, Karl E.; Rao, Duggirala K. M. (1997).Numerical Range: The Field of Values of Linear Operators and Matrices. Universitext. New York, NY: Springer.doi:10.1007/978-1-4613-8498-4.ISBN 978-0-387-94835-5.ISSN 0172-5939.

Papers

Toeplitz, Otto (1918)."Das algebraische Analogon zu einem Satze von Fejér"(PDF).Mathematische Zeitschrift (in German).2 (1–2):187–197.doi:10.1007/BF01212904.ISSN 0025-5874.
Hausdorff, Felix (1919). "Der Wertvorrat einer Bilinearform".Mathematische Zeitschrift (in German).3 (1):314–316.doi:10.1007/BF01292610.ISSN 0025-5874.
Choi, M.D.; Kribs, D.W.; Życzkowski (2006), "Quantum error correcting codes from the compression formalism",Rep. Math. Phys.,58 (1):77–91,arXiv:quant-ph/0511101,Bibcode:2006RpMP...58...77C,doi:10.1016/S0034-4877(06)80041-8,S2CID 119427312.
Dirr, G.; Helmkel, U.; Kleinsteuber, M.; Schulte-Herbrüggen, Th. (2006), "A new type of C-numerical range arising in quantum computing",Proc. Appl. Math. Mech.,6:711–712,doi:10.1002/pamm.200610336
Li, C.K. (1996), "A simple proof of the elliptical range theorem",Proc. Am. Math. Soc.,124 (7): 1985,doi:10.1090/S0002-9939-96-03307-2.
Keeler, Dennis S.; Rodman, Leiba; Spitkovsky, Ilya M. (1997), "The numerical range of 3 × 3 matrices",Linear Algebra and Its Applications,252 (1–3): 115,doi:10.1016/0024-3795(95)00674-5.
Johnson, Charles R. (1976)."Functional characterizations of the field of values and the convex hull of the spectrum"(PDF).Proceedings of the American Mathematical Society.61 (2). American Mathematical Society (AMS):201–204.doi:10.1090/s0002-9939-1976-0437555-3.ISSN 0002-9939.

References

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^""well-known" inequality for numerical radius of an operator".StackExchange.
^"Upper bound for norm of Hilbert space operator".StackExchange.
^"Inequalities for numerical radius of complex Hilbert space operator".StackExchange.
^Hilary Priestley."B4b hilbert spaces: extended synopses 9. Spectral theory"(PDF).In fact, ‖T‖ = max(−m_T , M_T) = w_T. This fails for non-self-adjoint operators, but w_T ≤ ‖T‖ ≤ 2w_T in the complex case.
^Davis, Chandler (June 1971)."The Toeplitz-Hausdorff Theorem Explained".Canadian Mathematical Bulletin.14 (2):245–246.doi:10.4153/CMB-1971-042-7.ISSN 0008-4395.
^Choi, Man-Duen; Kribs, David W.; Życzkowski, Karol (October 2006)."Higher-rank numerical ranges and compression problems".Linear Algebra and its Applications.418 (2–3):828–839.doi:10.1016/j.laa.2006.03.019.
^Li, Chi-Kwong; Sze, Nung-Sing (2008)."Canonical Forms, Higher Rank Numerical Ranges, Totally Isotropic Subspaces, and Matrix Equations".Proceedings of the American Mathematical Society.136 (9):3013–3023.ISSN 0002-9939.
^Woerdeman, Hugo J. (2008-01-01)."The higher rank numerical range is convex".Linear and Multilinear Algebra.56 (1–2):65–67.doi:10.1080/03081080701352211.ISSN 0308-1087.
^Li, Chi-Kwong; Poon, Yiu-Tung; Sze, Nung-Sing (2009-06-01)."Condition for the higher rank numerical range to be non-empty".Linear and Multilinear Algebra.57 (4):365–368.arXiv:0706.1540.doi:10.1080/03081080701786384.ISSN 0308-1087.

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Properties	Barrelled Complete Dual (Algebraic /Topological) Locally convex Reflexive Separable